October 2005 - mathfun@mailman.xmission.com

RE: [math-fun] batting averages
by dasimov＠earthlink.net 26 Oct '05

26 Oct '05

Michael Reid wrote: << [dan asimov wrote]: > Michael Reed wrote: who?? >> My apologies to you for misspelling your name. --Da3n

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Re: [math-fun] batting averages
by Phil Carmody 26 Oct '05

26 Oct '05

From: "R. William Gosper" > At minimum, > (c2) block([keepfloat : true],linsolve(2*n/(5*n+1) = 0.3985)) > > (d2) [n = 53.1321] > > (c3) ceiling(subst(%,5*n+1)) > > (d3) 267 > at bats. 55 from 138 is 0.39855. So is 138 the improbable number of 'at bats'? Disclaimer - I have no idea how the deviant version of rounders that Westpondians play works. Phil () ASCII ribbon campaign () Hopeless ribbon campaign /\ against HTML mail /\ against gratuitous bloodshed [stolen with permission from Daniel B. Cristofani] __________________________________ Yahoo! FareChase: Search multiple travel sites in one click. http://farechase.yahoo.com

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RE: [math-fun] basketball shooting problem
by Andy Latto 26 Oct '05

26 Oct '05

> -----Original Message----- > From: math-fun-bounces+andy.latto=pobox.com(a)mailman.xmission.com > [mailto:math-fun-bounces+andy.latto=pobox.com@mailman.xmission.com]On > Behalf Of dasimov(a)earthlink.net > Sent: Tuesday, October 25, 2005 9:49 PM > To: math-fun > Subject: RE: [math-fun] basketball shooting problem > > > Andy wrote, in conclusion: > > << > So it is impossible to cross the target without hitting it if > the target for hits/misses is an integer K, or equivalently, if > (hits/(hits + misses)) is of the form K/K+1 for some integer K. > >> > > Aha -- so by symmetry of hits <--> misses, this also implies the > impossibility of jumping over the target if it's of the form 1/(K+1). Careful! You have to reverse "above" and "below" when you exploit the symmetry. You can jump from below 1/(K+1) to above 1/(K+1), but you can't jump from above the target to below it without hitting it exactly if it's of the form 1/(K+1). For targets of the form K/(K+1), you can jump from above to below without hitting the target exactly, but must hit the target going from below to above. And those are the only "mandatory target hit" numbers. Andy Latto andy.latto(a)pobox.com >

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RE: [math-fun] basketball shooting problem
by dasimov＠earthlink.net 26 Oct '05

26 Oct '05

Andy wrote, in conclusion: << So it is impossible to cross the target without hitting it if the target for hits/misses is an integer K, or equivalently, if (hits/(hits + misses)) is of the form K/K+1 for some integer K. >> Aha -- so by symmetry of hits <--> misses, this also implies the impossibility of jumping over the target if it's of the form 1/(K+1). --Dan

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RE: [math-fun] batting averages
by Michael Reid 26 Oct '05

26 Oct '05

dan asimov: > Michael Reed wrote: who?? > << > . . .when i was a (younger!) kid in little league baseball, one of > my friends told me that he had calculated his batting average > . . . and that it was .399 . immediately i knew that this could not > be correct. how did i know? > >> > > I'm gonna guess that if |2/5 - p/q| < 1/1000 > then q must be greater than any reasonable > number of at-bats in one Little League season. this is essentially it, although the inequality should be .5 / 1000 < 2/5 - p/q < 1.5 / 1000 . fwiw, i remember wondering at the time if it was possible that my friend's calculator read ".39999999" and they truncated instead of rounding. this doesn't seem a reasonable explanation. > Andy wrote, in conclusion: > > << > So it is impossible to cross the target without hitting it if > the target for hits/misses is an integer K, or equivalently, if > (hits/(hits + misses)) is of the form K/K+1 for some integer K. > >> > > Aha -- so by symmetry of hits <--> misses, this also implies the > impossibility of jumping over the target if it's of the form 1/(K+1). > > --Dan it depends which direction one jumps over the target, see my earlier message. before-above and after-below requires one hits the target. before-below and after-above does not. mike

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RE: [math-fun] basketball shooting problem
by Andy Latto 25 Oct '05

25 Oct '05

> -----Original Message----- > From: math-fun-bounces+andy.latto=pobox.com(a)mailman.xmission.com > [mailto:math-fun-bounces+andy.latto=pobox.com@mailman.xmission.com]On > Behalf Of Michael Reid > Sent: Tuesday, October 25, 2005 1:00 PM > To: math-fun(a)mailman.xmission.com > Subject: RE: [math-fun] basketball shooting problem > > > the solutions in the monthly appear to strive for conciseness, > which sometimes yield less insight. I agree. Here is my solution, which I found simple enough to solve it mentally. Instead of considering the ratio (Hits/total throws), consider the ratio (Hits/misses). Since the latter is a strictly monotonic function of the latter, you can pass from below the target to above the target of one of these ratios exactly if you can do so with the other one. If the target (hits/misses) ratio is K, then the goal is to find integers M and H such that H is less than MK, but H+1 is greater than MK. It is clear that this is possible only if MK is not an interger, which in turn is only possible for some choice of M if K is not an integer. So it is impossible to cross the target without hitting it if the target for hits/misses is an integer K, or equivalently, if (hits/(hits + misses)) is of the form K/K+1 for some integer K. Moral: choosing the right parametrization makes things easier. Andy Latto andy.latto(a)pobox.com >

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[math-fun] batting averages
by Michael Reid 25 Oct '05

25 Oct '05

this reminds me of a story: when i was a (younger!) kid in little league baseball, one of my friends told me that he had calculated his batting average (actually, i think he said his dad helped him) and that it was .399 . immediately i knew that this could not be correct. how did i know? mike

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RE: [math-fun] basketball shooting problem
by Michael Reid 25 Oct '05

25 Oct '05

the solutions in the monthly appear to strive for conciseness, which sometimes yield less insight. i also originally found it to be counterintuitive, in that it felt as if there should be a p <---> 1 - p symmetry. in fact, there is, if one also switches "below" and "above" (or "before" and "after"). here's my solution, which i think offers some insight. suppose the target ratio is a/b in lowest terms. keep track of an auxiliary score, which is defined as b * (number of successes) - a * (number of attempts) we are given that at some point, this score is negative, and at a later point it is positive. each successful attempt adds b - a to the score, and each miss subtracts a . in the original problem, a = 4 and b = 5 , so the score either increases by 1 or decreases by 4 . now it's easy to see why you must hit 0 exactly when going from below 0 to above. if the positive step b - a is greater than 1 this also suggests how to get from below 0 to above without actually hitting 0 , and indeed this is possible (because a and b are relatively prime). mike

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RE: [math-fun] basketball shooting problem
by dasimov＠earthlink.net 25 Oct '05

25 Oct '05

Chris wrote: << . . . << Thane Plambeck asked Say you're shooting baskets and you keep track of your record as you shoot them . . . Someone specifies a "target ratio" (I think it was 80% in the problem statement) and you're told that during the record of shots, the shooter was below the target ratio, but later was better than the target ratio. Is there necessarily a moment at which the shooter must have shot exactly that ratio up to that point? >> so i now think that the answer is yes for 80% = 4/5 and any other a/(a+1) fraction, but no for any others >> I found the answer to this question (I think from the last Putnam exam) to be totally counterintuitive. (The target ratio was in fact 80%.) Here's the proof from the Monthly: As you come up from below 80% to greater than it, there is a last moment when you're still below 80%. Say that was when you'd made p-1 baskets out of q-1 tries; after the next shot you're greater than or equal to 80%. Therefore: p-1/q-1 < 4/5, implying 4q+1 > 5. And 4/5 <= p/q, implying 4q <= 5p The two inequalities guarantee that 4q = 5p. Amazing. --Dan

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[math-fun] basketball shooting problem
by Thane Plambeck 25 Oct '05

25 Oct '05

This is a problem from the last Putnam exam that had me scribbling (which I don't have in front of me, but no matter) Say you're shooting baskets and you keep track of your record as you shoot them, for example 0 for 1 1 for 2 1 for 3 1 for 4 1 for 5 2 for 6 etc, you get the idea Someone specifies a "target ratio" (I think it was 80% in the problem statement) and you're told that during the record of shots, the shooter was below the target ratio, but later was better than the target ratio. Is there necessarily a moment at which the shooter must have shot exactly that ratio up to that point? Answer this for 80% and then generalize -- Thane Plambeck http://www.plambeck.org/ehome.htm

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